On any given trial, the probability that a particular outcome will occur is
constant.

All of the trials in the experiment are independent.

Tossing a pair of dice is a perfect example of a multinomial
experiment. Suppose we toss a pair of dice three times. Each toss represents a
trial, so this experiment would have 3 trials. Each toss also has a discrete
number of possible outcomes - 2 through 12. The probability of any particular
outcome is constant; for example, the probability of rolling a 12 on any
particular toss is always 1/36. And finally, the outcome on any toss is not
affected by previous or succeeding tosses; so the trials in the experiment are
independent.

What is a multinomial distribution?

A multinomial distribution is a
probability distribution. It refers to the probabilities associated
with each of the possible outcomes in a multinomial experiment.

For example, suppose we toss a toss a pair of dice one time. This
multinomial experiment has 11 possible outcomes: the numbers from 2 to 12. The
probabilities associated with each possible outcome are an example of a
multinomial distribution, as shown below.

Outcome

2

3

4

5

6

7

8

9

10

11

12

Probability

1/36

2/36

3/36

4/36

5/36

6/36

5/36

4/36

3/36

2/36

1/36

The table completely defines the probabilities associated with
every possible outcome from this multinomial experiment. It is the multinomial
distribution for this experiment.

What is the number of outcomes?

The number of outcomes refers to the number of different results
that could occur from a multinomial experiment. For example, suppose we roll a
die. Each roll of the die can have six possible outcomes - 1, 2, 3, 4, 5, or 6.
Similarly, the roll of two dice can have eleven possible outcomes - the numbers
from 2 to 12.

What is the probability of an
outcome?

Each trial in a multinomial experiment can have discrete number
of outcomes. The likelihood that a particular outcome will occur in a single
trial is the probability of the outcome.

For example, suppose we toss two dice. The probability of tossing
a 2 is 1/36; the probability of tossing a 3 is 2/36, the probability of tossing
4 is 3/36, etc.

What is the frequency of an outcome?

In a multinomial experiment, the frequency of an outcome refers
to the number of times that an outcome occurs. For example, suppose we toss a
single die four times and observe the following outcomes: we roll a 1,
a 3, and a two 5's? The frequency for each outcome is shown in the table below.

Outcome

1

2

3

4

5

6

Frequency

1

0

1

0

2

0

What is the multinomial probability?

A multinomial probability refers to the probability of obtaining
a specified frequency in a multinomial experiment. For example, suppose we toss
a single die four times. We might ask: What is the probability that we roll a
1, a 3, and a two 5's?

A binomial experiment is actually a special case of a multinomial
experiment. The binomial experiment is a multinomial experiment, in which each
trial can have only two possible outcomes. The flip of a coin is a good example
of a binomial experiment, since a coin flip can have only two possible outcomes
- heads or tails. To learn more about binomial experiments, go to Stat Trek's
tutorial on the binomial distribution.

Multinomial Distribution: Sample Problems

Suppose you toss a pair of dice 10 times. What is the probability of getting
the following outcome: two rolls of 7, two rolls of 6, and any other outcome on
the remaining six rolls.

Hint: The probabilities associated with each roll of two dice are shown below.

Outcome

2

3

4

5

6

7

8

9

10

11

12

Probability

1/36

2/36

3/36

4/36

5/36

6/36

5/36

4/36

3/36

2/36

1/36

Solution:

We know the following:

The number of outcomes is 3. (Outcome 1 is a roll of 7, the Outcome 2 is a roll
of 6, and Outcome 3 is any other roll.)

Outcome 1: The probability is 6/36, and the frequency is 2.

Outcome 2: The probability is 5/36, and the frequency is 2.

Outcome 3: The probability is 25/36, and the frequency is 6.

Therefore, we plug those numbers into the Multinomial
Calculator
and hit the Calculate button. The calculator reports that the multinomial
probability is 0.076. Thus, in ten rolls of the dice, the probability of
rolling 7 two times, 6 two times, and something else six times
is 0.076.

An bowl has 2 black marbles, 3 green marbles, and 5 white marbles. A marble is
randomly selected and then put back in the bowl. Suppose this selection process
is repeated five times. What is the probability that 3 white marbles, 1 green
marble, and 1 black marble will be chosen?

Hint: On any given trial, the probability of choosing a black marble is 2/10;
the probability of choosing a green marble is 3/10; and
the probability of choosing a white marble is 5/10.

Solution:

We know the following:

The number of outcomes is 3. (Outcome 1 is a black marble; Outcome 2, a green
marble; and Outcome 3, a white marble.)

Outcome 1: The probability is 0.2, and the frequency is 1.

Outcome 2: The probability is 0.3, and the frequency is 1.

Outcome 3: The probability is 0.5, and the frequency is 3.

Therefore, we plug those numbers into the Multinomial
Calculator and hit the Calculate button. The calculator reports that
the cumulative multinomial probability is 0.150. Thus, the probability of
selecting 1 black marble, 1 green marble, and 3 white marbles is 0.150.